Completeness of the L1-space of closed vector measures

1990 ◽  
Vol 33 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Werner J. Ricker
Keyword(s):  
Uniform Convergence ◽  
Convex Space ◽  
Vector Measure ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Vector Measures ◽  
Locally Convex

The notion of a closed vector measure m, due to I. Kluv´;nek, is by now well established. Its importance stems from the fact that if the locally convex space X in which m assumes its values is sequentially complete, then m is closed if and only if its L1-space is complete for the topology of uniform convergence of indefinite integrals. However, there are important examples of X-valued measures where X is not sequentially complete. Sufficient conditions guaranteeing the completeness of L1(m) for closed X-valued measures m are presented without the requirement that X be sequentially complete.

scholarly journals Injective absolutely summing operators

1988 ◽  
Vol 103 (3) ◽  
pp. 497-502
Author(s):  
Susumu Okada ◽  
Yoshiaki Okazaki
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Space ◽  
Locally Convex Space ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Integrable Functions ◽  
Locally Convex ◽  
Convergence In Mean

Let X be an infinite-dimensional Banach space. It is well-known that the space of X-valued, Pettis integrable functions is not always complete with respect to the topology of convergence in mean, that is, the uniform convergence of indefinite integrals (see [14]). The Archimedes integral introduced in [9] does not suffer from this defect. For the Archimedes integral, functions to be integrated are allowed to take values in a locally convex space Y larger than the space X while X accommodates the values of indefinite integrals. Moreover, there exists a locally convex space Y, into which X is continuously embedded, such that the space ℒ(μX, Y) of Y-valued, Archimedes integrable functions is identical to the completion of the space of X valued, simple functions with repect to the toplogy of convergence in mean, for each non-negative measure μ (see [9]).

scholarly journals The bounded vector measure associated to a conical measure and pettis differentiability

2001 ◽  
Vol 70 (1) ◽  
pp. 10-36
Author(s):  
L. Rodriguez-Piazza ◽  
M. C. Romero-Moreno
Keyword(s):  
Vector Measure ◽  
Locally Convex Space ◽  
Finite Intersection ◽  
Vector Measures ◽  
Weak Space ◽  
Countably Additive Vector Measure ◽  
Differentiable Vector ◽  
Locally Convex ◽  
Additive Vector ◽  
Additive Vector Measure

AbstractLet X be a locally convex space. Kluvánek associated to each X-valued countably additive vector measure a conical measure on X; this can also be done for finitely additive bounded vector measures. We prove that every conical measure u on X, whose associated zonoform Ku is contained in X, is associated to a bounded additive vector measure σ(u) defined on X, and satisfying σ(u)(H) ∈ H, for every finite intersection H of closed half-spaces. When X is a complete weak space, we prove that σ(u) is countably additive. This allows us to recover two results of Kluvánek: for any X, every conical measure u on it with Ku ⊆ X is associated to a countably additive X-valued vector measure; and every conical measure on a complete weak space is localizable. When X is a Banach space, we prove that σ(u) is countably additive if and only if u is the conical measure associated to a Pettis differentiable vector measure.

Necessary and Sufficient Conditions for the Equality of L(f) and l1

1982 ◽  
Vol 34 (2) ◽  
pp. 406-410 ◽  
Author(s):  
Waleed Deeb
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Locally Convex Space ◽  
Natural Question ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Infinite Dimensional Subspace

Introduction. Let f be a modulus, ei = (δij) and E = {ei, i = 1, 2, …}. The L(f) spaces were created (to the best of our knowledge) by W. Ruckle in [2] in order to construct an example to answer a question of A. Wilansky. It turned out that these spaces are interesting spaces. For example lp, 0 < p ≦ 1 is an L(f) space with f(x) = xp, and every FK space contains an L(f) space [2]. A natural question is: For which f is L(f) a locally convex space? It is known that L(f) ⊆ l1, for all f modulus (see [2]), and l1 is the smallest locally convex FK space in which E is bounded (see [1]). Thus the question becomes: For which f does L(f) equal l1? In this paper we characterize such f. (An FK space need not be locally convex here.) We also characterize those f for which L(f) contains a convex ball. The final result of this paper is to show that if f satisfies f(x · y) ≦ f(x) · f(y) and L(f) ≠ l1 then L(f) contains no infinite dimensional subspace isomorphic to a Banach space.

An ordered suprabarrelled space

1991 ◽  
Vol 51 (1) ◽  
pp. 106-117
Author(s):  
J. C. Ferrando ◽  
M. López-Pellicer
Keyword(s):  
Convex Space ◽  
Affirmative Answer ◽  
Locally Convex Space ◽  
Vector Measures ◽  
Locally Convex

AbstractA locally convex space E is said to be ordered suprabarrelled if given any increasing sequence of subspaces of E covering E there is one of them which is suprabarrelled. In this paper we show that the space m0(X, Σ), where X is any set and Σ is a σ-algebra on X, is ordered suprabarrelled, given an affirmative answer to a previously raised question. We also include two applications of this result to the theory of vector measures.

The Schwartz property and nuclearity of spaces of smooth and holomorphic functions in infinite dimensions

1990 ◽  
Vol 107 (2) ◽  
pp. 377-385
Author(s):  
Sten Bjon
Keyword(s):  
Uniform Convergence ◽  
Open Subset ◽  
Convex Space ◽  
Holomorphic Functions ◽  
Locally Convex Space ◽  
Infinite Dimensions ◽  
Continuous Convergence ◽  
Convergence Structure ◽  
Locally Convex ◽  
Local Uniform Convergence

In [8] it was shown that a locally convex space E is a Schwartz space if and only if the convergence algebras Hc(U) and He(U) of holomorphic functions on an open subset of E coincide, i.e. if and only if continuous convergence c (see [1]) and the associated equable convergence structure e (= local uniform convergence, see [2, 13]) coincide.

scholarly journals Regularization of cylindrical processes in locally convex spaces

2020 ◽  
pp. 2050027
Author(s):  
Christian A. Fonseca-Mora
Keyword(s):  
Convex Space ◽  
Lévy Process ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Levy Process ◽  
Locally Convex Spaces ◽  
Locally Convex ◽  
Strong Dual ◽  
Convex Spaces

Let [Formula: see text] be a locally convex space and let [Formula: see text] denote its strong dual. In this paper, we introduce sufficient conditions for the existence of a continuous or a càdlàg [Formula: see text]-valued version to a cylindrical process defined on [Formula: see text]. Our result generalizes many other known results on the literature and their different connections will be discussed. As an application, we provide sufficient conditions for the existence of a [Formula: see text]-valued càdlàg Lévy process version to a given cylindrical Lévy process in [Formula: see text].

scholarly journals Some properties of vector measures taking values in a topological vector space

1987 ◽  
Vol 43 (2) ◽  
pp. 224-230
Author(s):  
Efstathios Giannakoulias
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Necessary Condition ◽  
Vector Measures ◽  
Topological Vector Spaces ◽  
Locally Convex ◽  
Pettis Integrability

AbstractIn this paper we study some properties of vector measures with values in various topological vector spaces. As a matter of fact, we give a necessary condition implying the Pettis integrability of a function f: S → E, where S is a set and E a locally convex space. Furthermore, we prove an iff condition under which (Q, E) has the Pettis property, for an algebra Q and a sequentially complete topological vector space E. An approximating theorem concerning vector measures taking values in a Fréchet space is also given.

scholarly journals Bounded vector measures on effect algebras

2005 ◽  
Vol 72 (2) ◽  
pp. 291-298 ◽  
Author(s):  
Hong Taek Hwang ◽  
Longlu Li ◽  
Hunnam Kim
Keyword(s):  
Convex Space ◽  
Effect Algebra ◽  
Locally Convex Space ◽  
Effect Algebras ◽  
Vector Measures ◽  
Orthogonal Sequence ◽  
Locally Convex

Let (L, ⊥, ⊕,0,1) be an effect algebra and X a locally convex space with dual X′. A function μ: L → X is called a measure if μ(a ⊕ b) = μ(a) + μ(b) whenever a⊥b in L and it is bounded if is bounded for each orthogonal sequence {an} in L. We establish five useful conditions that are equivalent to boundedness for vector measures on effect algebras.

Egorov measurability and generator cores

1986 ◽  
Vol 100 (1) ◽  
pp. 137-143
Author(s):  
Brian Jefferies
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Unbounded Operators ◽  
Locally Convex

AbstractSufficient conditions are given for a set to be a core for the generator of a weakly integrable semigroup on a locally convex space. The conditions are illustrated by semigroups of unbounded operators on a Banach space.

scholarly journals A maximum principle

1979 ◽  
Vol 28 (1) ◽  
pp. 23-26
Author(s):  
Kung-Fu Ng
Keyword(s):  
Maximum Principle ◽  
Extreme Point ◽  
Convex Space ◽  
Maximal Element ◽  
Locally Convex Space ◽  
Compact Set ◽  
Natural Manner ◽  
Upper Semicontinuous ◽  
Nonempty Compact ◽  
Locally Convex

AbstractLet K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

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